Affine Lie algebras and Hecke modular forms

The character of a highest weight representation of an affine lie algebra can be written as a finite sum of products of classical 0-functions and certain modular functions, called string functions. We find the transformation law for the string functions, which allows us to compute them explicitly in many interesting cases. Finally, we write an explicit formula for the partition function, in the simplest case A[ x \ and compute the string functions directly. After multiplication by the cube of the T?-function, they turn out to be Hecke modular forms! 1. (See [3] or [7] for details.) Let g be a complex finite-dimensional simple lie algebra, a Cartan subalgebra of g. A the set of roots of in g. A + a set of positive roots, II = {OL X ,..., a ; } the corresponding set of simple roots, 0 the highest root. Let (,) be an invariant symmetric bilinear form on g normalized by (6,6) = 2. For a^* with (a, a) * 0 define H a G by 0(# a ) = 2(0, a)/(a, a) for j8 G *. Let W be the Weyl group of in g. Denote by M the Z-span of W6 (long roots). Let C[t, t" 1 ] be the algebra of Laurent polynomials over C in an indeterminate t. We regard g' := C[t, t~x] c g as an (infinite-dimensional) complex lie algebra. Define the affine Lie algebra g as follows. Let g = 'g Cc Cd and define the bracket by ( dx \ dx for x, y G 'gf. The algebra g is an important example of a Kac-Moody algebra [5], [10]. Note that Cc is the center of the algebra g. The subalgebra % = i) <B Cc 0 Giis called the Cartan subalgebra of g. For a G * set g a = {x G g | [ft, x] = a(h)x for ft G }; then we have the root space decomposition 8 = 9<r Detine a nondegenerate symmetric bilinear form (,) on by (ft, ft') is unchanged if ft, ft' G C 6, (ft, c) = (ft, d) = 0 for ft G fc>, (c, c) = (tf, J) = 0, (c, rf) = 1. We identify with * by this form; then * is identified with a subspace in %* by a(c) = a(<2) = 0 for a G *. For a G * set a = aL " so that Se * C |*. Define 6 G $* by 6(A) = 0 for ft G , 6(c) = 0, 6(d) = 1.